There are two methods to solve this question.
Method 1:Since x, y, z are all positive numbers, we can easily carry their denominators over the inequality without the fear of messing with the inequality.
Hence, \(\frac{x}{y} > \frac{x + z}{y + z}\) can be written as x(y+z)>y(x+z)
This implies xy+xz>xy+yz
Again, xy is a positive number here, so we cancel it out from both sides, leaving us with xz>yz. Taking the expression to one side of inequality, we get xz-yz>0
Thus, either
z>0 and x>y or
z<0 and x<y.
Now, we are given that z>0 (its a positive number). Hence, for the expression to be definitely true, we must also get x>y. And statement 2 does that for us.
Hence statement 2 is sufficientMethod 2:Now, if we add the same positive number z to x and y in the fraction x/y, \(\frac{x + z}{y + z}\) will come closer to 1.
This gives us two possibilities:
1) If x/y<1, then \(\frac{(x+z)}{(y+z)}\) is greater than x/y
and
2) If x/y>1, then \(\frac{(x+z)}{(y+z)}\) is smaller than x/y
In our case, 2) works and this is satisfied only by statement 2. Hence, sufficient
Answer is BNote: For anyone looking for more explanation on concept I used in statement 2, there is an article I think written by Karishma somewhere on GMATclub explaining this. If interested, I can look it up.
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